CMC hypersurface with finite index in hyperbolic space $\mathbb{H}^4$
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908405288927232 |
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| author | Hong, Han |
| author_facet | Hong, Han |
| contents | In this paper, we prove that there are no complete noncompact constant mean curvature hypersurfaces with the mean curvature $H > 1$, finite index and finite topology in hyperbolic space $\mathbb{H}^4$. A more general nonexistence result can be proved in a $4$-dimensional Riemannian manifold with certain curvature conditions. We also show that $4$-manifold with $\operatorname{Ric} > 1$ does not contain any complete noncompact minimal stable hypersurface with finite topology. The proof relies on the $μ$-bubble initially introduced by Gromov and further developed by Chodosh-Li-Stryker in the context of stable minimal hypersurfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_10276 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | CMC hypersurface with finite index in hyperbolic space $\mathbb{H}^4$ Hong, Han Differential Geometry 53A10, 49Q10 In this paper, we prove that there are no complete noncompact constant mean curvature hypersurfaces with the mean curvature $H > 1$, finite index and finite topology in hyperbolic space $\mathbb{H}^4$. A more general nonexistence result can be proved in a $4$-dimensional Riemannian manifold with certain curvature conditions. We also show that $4$-manifold with $\operatorname{Ric} > 1$ does not contain any complete noncompact minimal stable hypersurface with finite topology. The proof relies on the $μ$-bubble initially introduced by Gromov and further developed by Chodosh-Li-Stryker in the context of stable minimal hypersurfaces. |
| title | CMC hypersurface with finite index in hyperbolic space $\mathbb{H}^4$ |
| topic | Differential Geometry 53A10, 49Q10 |
| url | https://arxiv.org/abs/2404.10276 |